Optimal. Leaf size=16 \[ \frac{\sqrt{x^4+x^2+1}}{x} \]
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Rubi [A] time = 0.0180566, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {1590} \[ \frac{\sqrt{x^4+x^2+1}}{x} \]
Antiderivative was successfully verified.
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Rule 1590
Rubi steps
\begin{align*} \int \frac{-1+x^4}{x^2 \sqrt{1+x^2+x^4}} \, dx &=\frac{\sqrt{1+x^2+x^4}}{x}\\ \end{align*}
Mathematica [A] time = 0.0360015, size = 16, normalized size = 1. \[ \frac{\sqrt{x^4+x^2+1}}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 29, normalized size = 1.8 \begin{align*}{\frac{ \left ({x}^{2}+x+1 \right ) \left ({x}^{2}-x+1 \right ) }{x}{\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15252, size = 30, normalized size = 1.88 \begin{align*} \frac{\sqrt{x^{2} + x + 1} \sqrt{x^{2} - x + 1}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49778, size = 31, normalized size = 1.94 \begin{align*} \frac{\sqrt{x^{4} + x^{2} + 1}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{2} \sqrt{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} - 1}{\sqrt{x^{4} + x^{2} + 1} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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